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Comments to the question (v2):

1. Recall that the Lorentz group $G=O(3,1)$ has 4 connected components $$ G~=~G_0 ~\cup~ P\cdot G_0 ~\cup~ T\cdot G_0 ~\cup~ PT\cdot G_0. $$ Here the connected components $G_0$ that contains the identity is the restricted Lorentz group $G_0=SO^+(3,1)$.

2. It is straightforward to see that $G_0$ is a normal subgroup of $G$. Therefore the quotient $G/G_0$ is a group.

3. For the Lorentz group $G=O(3,1)$, the quotient $G/G_0$ is isomorphic to the Klein Vierergruppe $V\cong \mathbb{Z}_2\times \mathbb{Z}_2$, which is generated by two elements.

4. Moreover, the quotient $G/G_0$ is discrete. Formally speaking, the elements of $G/G_0$ constitute the 'discrete Lorentz transformations'. In practice, one often picks a representative for each equivalence class, and called these representatives the 'discrete Lorentz transformations', with the implicit understanding that one might as well pick other representatives, that deviate with elements of $G_0$.

5. Returning to OP's example, the difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, which belongs to $G_0$, cf. above comment by Meng Cheng.

1. Recall that the Lorentz group $G=O(3,1)$ has 4 connected components $$ G~=~G_0 ~\cup~ P\cdot G_0 ~\cup~ T\cdot G_0 ~\cup~ PT\cdot G_0. $$ Here the connected components $G_0$ that contains the identity is the restricted Lorentz group $G_0=SO^+(3,1)$.

2. It is straightforward to see that $G_0$ is a normal subgroup of $G$. Therefore the quotient $G/G_0$ is a group.

3. For the Lorentz group $G=O(3,1)$, the quotient $G/G_0$ is isomorphic to the Klein Vierergruppe $V\cong \mathbb{Z}_2\times \mathbb{Z}_2$, which is generated by two elements.

4. Moreover, the quotient $G/G_0$ is discrete. Formally speaking, the elements of $G/G_0$ constitute the 'discrete Lorentz transformations'. In practice, one often picks a representative for each equivalence class, and called these representatives the 'discrete Lorentz transformations', with the implicit understanding that one might as well pick other representatives, that deviate with elements of $G_0$.

5. Returning to OP's example, the difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, which belongs to $G_0$, cf. above comment by Meng Cheng.

Comments to the question (v2):

1. Recall that the Lorentz group $G=O(3,1)$ has 4 connected components $$ G~=~G_0 ~\cup~ P\cdot G_0 ~\cup~ T\cdot G_0 ~\cup~ PT\cdot G_0. $$ Here the connected components $G_0$ that contains the identity is the restricted Lorentz group $G_0=SO^+(3,1)$.

2. It is straightforward to see that $G_0$ is a normal subgroup of $G$. Therefore the quotient $G/G_0$ is a group.

3. For the Lorentz group $G=O(3,1)$, the quotient $G/G_0$ is isomorphic to the Klein vier-group $\mathbb{Z}_2\times \mathbb{Z}_2$, which is generated by two elements.

4. Moreover, the quotient $G/G_0$ is discrete. Formally speaking, the elements of $G/G_0$ constitute the 'discrete Lorentz transformations'. In practice, one often picks a representative for each equivalence class, and called these representatives the 'discrete Lorentz transformations', with the implicit understanding that one might as well pick other representatives, that deviate with elements of $G_0$.

5. Returning to OP's example, the difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, which belongs to $G_0$, cf. above comment by Meng Cheng.

Comments to the question (v2):

1. Recall that the Lorentz group $G=O(3,1)$ has 4 connected components $$ G~=~G_0 ~\cup~ P\cdot G_0 ~\cup~ T\cdot G_0 ~\cup~ PT\cdot G_0. $$ Here the connected components $G_0$ that contains the identity is the restricted Lorentz group $G_0=SO^+(3,1)$.

2. It is straightforward to see that $G_0$ is a normal subgroup of $G$. Therefore the quotient $G/G_0$ is a group.

3. For the Lorentz group $G=O(3,1)$, the quotient $G/G_0$ is isomorphic to the Klein Vierergruppe $V\cong \mathbb{Z}_2\times \mathbb{Z}_2$, which is generated by two elements.

4. Moreover, the quotient $G/G_0$ is discrete. Formally speaking, the elements of $G/G_0$ constitute the 'discrete Lorentz transformations'. In practice, one often picks a representative for each equivalence class, and called these representatives the 'discrete Lorentz transformations', with the implicit understanding that one might as well pick other representatives, that deviate with elements of $G_0$.

5. Returning to OP's example, the difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, which belongs to $G_0$, cf. above comment by Meng Cheng.

Comments to the question (v1):

1. The difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, cf. above comment by Meng Cheng.

2. Recall that the Lorentz group $G=O(3,1)$ has 4 connected components $$ G~=~G_0 ~\cup~ P\cdot G_0 ~\cup~ T\cdot G_0 ~\cup~ PT\cdot G_0. $$ The connected components $G_0$ that contains the identity is the restricted Lorentz group $G_0=SO^+(3,1)$.

3. It is straightforward to see that $G_0$ is a normal subgroup of $G$. Therefore the quotient $G/G_0$ is a group.

4. For the Lorentz group $G=O(3,1)$, the quotient $G/G_0$ is discrete. Formally, the elements of $G/G_0$ constitute the discrete Lorentz transformations. In practice, one often picks a representative for each equivalence class, and called them the discrete Lorentz transformations.

5. The quotient $G/G_0$ is isomorphic to the Klein vier-group $\mathbb{Z}_2\times \mathbb{Z}_2$.

Comments to the question (v2):

1. Recall that the Lorentz group $G=O(3,1)$ has 4 connected components $$ G~=~G_0 ~\cup~ P\cdot G_0 ~\cup~ T\cdot G_0 ~\cup~ PT\cdot G_0. $$ Here the connected components $G_0$ that contains the identity is the restricted Lorentz group $G_0=SO^+(3,1)$.

2. It is straightforward to see that $G_0$ is a normal subgroup of $G$. Therefore the quotient $G/G_0$ is a group.

3. For the Lorentz group $G=O(3,1)$, the quotient $G/G_0$ is isomorphic to the Klein vier-group $\mathbb{Z}_2\times \mathbb{Z}_2$, which is generated by two elements.

4. Moreover, the quotient $G/G_0$ is discrete. Formally speaking, the elements of $G/G_0$ constitute the 'discrete Lorentz transformations'. In practice, one often picks a representative for each equivalence class, and called these representatives the 'discrete Lorentz transformations', with the implicit understanding that one might as well pick other representatives, that deviate with elements of $G_0$.

5. Returning to OP's example, the difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, which belongs to $G_0$, cf. above comment by Meng Cheng.